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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>If <span class="process-math">\(\lambda&gt;0\text{,}\)</span> we write <span class="process-math">\(\lambda=k^2\text{,}\)</span> where <span class="process-math">\(k=\sqrt{|\lambda|}&gt;0\text{,}\)</span> then <span class="process-math">\(r^2 =-k^2~\to~r=\pm ki\text{,}\)</span> and</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y(x)=c_1\cos {kx} + c_2\sin {kx}.
\end{equation*}
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<p class="continuation">By setting in boundary conditions:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\left\{\begin{array}{l} y(0)={c_1~=0}\\
y(L)=c_2\sin kL=0\end{array}\right.
\end{equation*}
</div>
<p class="continuation">So it must be <span class="process-math">\(c_2\neq 0\)</span> and <span class="process-math">\(\sin kL = 0\)</span> leads to</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
kL=n\pi,\quad \Rightarrow \quad k=\frac{n\pi}{L},~~n=1,2,3,\cdots
\end{equation*}
</div>
<p class="continuation">We have found a family (infinite size) of eigenvalues and eigenfunctions! Using <span class="process-math">\(n\)</span> as the index, they are</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\lambda_n}={\left(\frac{n\pi}{L}\right)^2},\quad {y_n(x)}={\sin\frac{n\pi x}{L}},\quad n={1},2,3,\cdots
\end{equation*}
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